Question

1- If four alphabets are to be chosen from L, M, N, O, P, Q such that repetition is not allowed then in how many ways it can be done?

2- Sally must choose a 5-digit PIN. Each digit can be chosen from 0 to 9 and the same digit can be used repeatedly. How many possible PIN numbers can be created?

3- Jenny has 8 tops, 5 bottoms, 2 belts, and 2 bracelets. Assuming that everything matches, how many different outfits can Jenny create?

4- At the local pizza show a patron can choose either a large or medium size pizza, one of 5 different toppings and one of 3 types of meat. How many different types of pizza can be ordered?

5- A French restaurant offers a menu consisting of 4 different appetizers, 5 different soups, 3 different salads, 10 different main courses, and 7 different desserts.

A dinner special consists of a choice of soup or salad, plus a main course. How many dinner specials are possible?

6- A French restaurant offers a menu consisting of 3 different appetizers, 2 different soups, 7 different salads, 10 different main courses, and 6 different desserts.

A fixed-price lunch meal consists of a choice of appetizer, salad, and main course. How many different fixed-price lunch meals are possible?

7- A password consists of two letters of the alphabet followed by 4 digits chosen from 0 to 9. Repeats are not allowed. How many possible passwords can be created?

8- How many words of 4 letters can be created that start with a vowel and end with the letter Q? Note that a word is any combination of 4 capital letters and letters can be repeated.

Answer 1

Problem (1)

You have to choose four letters from L, M, N, O, P, Q without repetition.

The first letter can be chosen in 6 ways since there are six choices.

Once the first letter has been chosen, there are five choices left since repetition is not allowed.

So, the second letter can be chosen in 5 ways.

Once the second letter has been chosen, there are four choices left.

So, the third letter can be chosen in 4 ways.

Once the third letter has been chosen, there are three choices left.

So, the fourth letter can be chosen in 3 ways.

So, by the principle of counting, total number of ways = 6 $\times$ 5 $\times$ 4 $\times$ 3 = 360

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Problem (2)

Sally must choose 5 digits from 0, 1, 2, ..., 9 repetition allowed.

The first digit can be chosen in 10 ways since there are ten choices.

Once the first digit has been chosen, there are still ten choices left since repetition is allowed.

So, the second digit can be chosen in 10 ways.

Once the second digit has been chosen, there are still ten choices left.

So, the third letter can be chosen in 10 ways. And so on.

Going this way, there are 10 choices for each of the five digits.

So, by the principle of counting, total number of PINs = 10 $\times$ 10 $\times$ 10 $\times$ 10 $\times$ 10 = 100000

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Problem (3)

An outfit is made up of one top, one bottom, one belt, and one bracelet.

Since everything matches, she can wear any of them.

So, she has to choose one top, one bottom, one belt, and one bracelet from 8 tops, 5 bottoms, 2 belts, and 2 bracelets respectively.

This can be done in 8 $\times$ 5 $\times$ 2 $\times$ 2 = 160 ways

So, she can create 160 different outfits.

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Problem (4)

To order a pizza, you have three choices to make:

(1) Choose a size. You have 2 options on size. This can be done in 2 ways.

(2) Choose one topping. You have 5 options. This can be done in 5 ways.

(3) Choose a type of meat: You have 3 options. This can be done in 3 ways.

So, by the principle of counting, total number of pizzas that can be ordered = 2 $\times$ 5 $\times$ 3 = 30

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Problem (5)

You have two choices to make:

(1) Choose a soup or salad. You have 5+3 = 8 options. This can be done in 8 ways.

(2) Choose a main course. You have 10 options. This can be done in 10 ways.

So, by the principle of counting, total number of dinner specials = 8 $\times$ 10 = 80

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Problem (6)

You have three choices to make:

(1) Choose an appetizer. You have 3 options. This can be done in 3 ways.

(2) Choose a salad. You have 7 options. This can be done in 7 ways.

(3) Choose a main course. You have 10 options. This can be done in 10 ways.

So, by the principle of counting, total number of different fixed-price lunch meals = 3 $\times$ 7 $\times$ 10 = 210

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Problem (7)

There are 26 letters in the alphabet.

The first character is a letter. So, it can be chosen in 26 ways.

Once the first character (letter) has been chosen, there are 25 letters left since repetition is not allowed.

The second character is also a letter. So, it can be chosen in 25 ways.

Now, the third character is a digit. So, it can be chosen in 10 ways from 0, 1, 2, ..., 9.

Once the third character (digit) has been chosen, there are 9 digits left since repetition is not allowed.

The fourth character is also a digit. So, it can be chosen in 9 ways.

Once the fourth character (digit) has been chosen, there are 8 digits left.

The fifth character is also a digit. So, it can be chosen in 8 ways.

Once the fifth character (digit) has been chosen, there are 7 digits left.

The sixth character is also a digit. So, it can be chosen in 7 ways.

So, by the principle of counting, total number of passwords = 26 $\times$ 25 $\times$ 10 $\times$ 9 $\times$ 8 $\times$ 7 = 3276000

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Problem (8)

How many words of 4 letters can be created that start with a vowel and end with the letter Q?

The last letter is fixed as Q. You only have to choose the first three letters.

The first letter is a vowel. There are total five vowels. So, it can be done in 5 ways.

Once the first letter has been chosen, there are still 26 letters left since repetition is allowed.

So, the second letter can be chosen in 26 ways.

Once the second letter has been chosen, there are still 26 letters left.

So, the third letter can be chosen in 26 ways.

So, by the principle of counting, total number of words = 5 $\times$ 26 $\times$ 26 = 3380

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